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Hyperovals in Knuth's Binary Semifield Planes

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Hyperovals in Knuth's Binary Semifield Planes HYPEROVALS IN KNUTH’S BINARY SEMIFIELD PLANES NICOLA DURANTE, ROCCO TROMBETTI, AND YUE ZHOU Abstract. In each of the three projective planes coordinatised by the Knuth’s n binary semifield Kn of order 2 and two of its Knuth derivatives, we exhibit a new family of infinitely many translation hyperovals. In particular, when n = 5, we also present complete lists of all translation hyperovals in them. The properties of some designs associated with these hyperovals are also studied. 1. Introduction A semifield S is an algebraic structure satisfying all the axioms of a skewfield except (possibly) the associativity. A finite field is a trivial example of a semifield. Furthermore, if S does not necessarily have a multiplicative identity, then it is called a presemifield. In particular, when the multiplication of S is commutative, we also say that S is commutative. For a presemifield S, its additive group (S, +) is necessarily abelian [23]. The multiplication of S is not necessarily commutative or associative. However, by Wed- derburn’s Theorem, in the finite case, associativity implies commutativity. There- fore, a non-associative finite commutative semifield is the closest algebraic structure to a finite field. Geometrically speaking, there is a well-known correspondence, via coordinati- sation, between (pre)semifields and projective planes of Lenz-Barlotti type V.1, see [11, 17]. In [1], Albert showed that two (pre)semifields coordinatise isomorphic planes if and only if they are isotopic. Any presemifield can always be “normalized” into a semifield under a certain isotopism, which implies that there is no essential difference between semifields and presemifields. An isotopism from a semifield to itself is called an autotopism, and all autotopisms of a semifield form a group. We refer to [25] for a recent and comprehensive survey on semifields. In [22], Knuth presented a family of binary presemifields defined over F2mn where arXiv:1605.06267v1 [math.CO] 20 May 2016 n is odd. The multiplication in it is defined by 2 x y := xy + yTr mn m (x)+ xTr mn m (y) , ∗ 2 /2 2 /2 F F where Tr2mn/2m ( ) is the trace function from 2mn to 2m . It is clear that the multiplication x ·y is commutative, i.e. x y = y x. In this paper, we concentrate ∗ ∗ ∗ on the Knuth’s binary presemifields with m = 1 and denote them by Kn. For convenience, we write Tr2n/2(x) as Tr(x). Let Π be a finite projective plane of order q. An oval is a set of q + 1 points, no three of which are collinear. Due to Segre’s theorem [36], every oval in PG(2, q) with q odd is equivalent to a nondegenerate conic in the plane. When q is even, every oval can be uniquely extended to a set of q +2 points, no three of which are still collinear. We call this set of q + 2 points a hyperoval. Equivalently, we can also Date: September 13, 2018. 1 2 N.DURANTE,R.TROMBETTI,ANDY.ZHOU say a hyperoval is a maximal arc of degree 2. Dually, a line hyperoval is a set of q + 2 lines, no three of which are concurrent. From the definition of hyperovals, it is easy to show that each line in Π meets a hyperoval at either 0 or 2 points, and the line is called exterior or secant, respec- tively. For two different hyperovals and in Π, we say that they are equivalent O1 O2 if there is a collineation ϕ of the plane such that ϕ( 1)= 2. When q is even, a conic together with its nucleusO in PG(2O, q) is a regular hyper- oval (also known as a complete conic). Different from the q odd case, for q even, there are several inequivalent hyperovals in PG(2, q) besides the regular hyperovals. Finding these hyperovals has been a hard work of almost 40 years and a complete classification of hyperovals in PG(2, q) seems elusive. We refer to [8] and [6, 15] for a list of known hyperovals in PG(2, q) and recent classification results, respectively. Let Π be a translation plane of even order q with a translation line l . A hyperoval is called a translation hyperoval, if ∞ O the line l is secant to , • there exists∞ a subgroupO of order q in the translation group acting regularly • on the q affine points of . O The set l is also known as the carrier set of the hyperoval. When Π is PG(2, q) and∞q∩=2 On, without loss of generality, we can always assume that ( ), (0) and (0, 0) are points of the hyperoval and the affine points of can be∞ written as O O (x, f(x)) : x F . { ∈ q} In [16, 33], it has been shown that a function f defines a translation hyperoval k in PG(2, q) if and only if f(x) = x2 where gcd(n, k) = 1. It implies a complete classification of translation hyperovals in desarguesian planes. There are a few known hyperovals in non-desarguesian planes of even orders, which were discovered approximately 20 years ago. In generalized Andr´eplanes, there are translation hyperovals inherited from those in desarguesian planes; see [12, 18]. In a Hall plane of order q2, the affine points of certain conics in PG(2, q2) can also be extended into hyperovals; see [24, 32]. In finite Figueroa planes of even order, there are hyperovals inherited from regular ones in the associated desarguesian planes; see [10]. A complete list of hyperovals in all projective planes of order 16 is presented in [34], in which it is also shown that there do exist projective planes containing no hyperovals. In this paper, we present three new families of translation hyperovals; one in the semifield plane Π(Kn) coordinated by the Knuth’s binary presemifield Kn = t td (F n , +, ), one in Π(K ) and another one in Π(K ), where t and d are the transpose 2 ∗ n n and the dual operation on Kn, respectively. In particular, in each relevant plane we give a complete list of translation hyperovals, when n = 5. Throughout this paper, we denote the points and the lines of a semifield plane Π(S) in the following way: The set of affine points in Π(S) is (a,b): a,b Fq . There are another q +1 points on the line l at infinity, and we denote{ them by∈ (a}), where a F . Symmetrically, all the∞ q + 1 lines through ( ) are denoted ∈ q ∪ {∞} ∞ by la, where a Fq . The affine points on la are (a,y) for all y Fq. For any a,b F , the∈ line∪l {∞}is defined by ∈ ∈ q a,b l := (x, y): y = x⋆a + b (a) , a,b { }∪{ } where ⋆ is the multiplication in S. HYPEROVALS IN KNUTH’S BINARY SEMIFIELD PLANES 3 The rest of this paper is organized as follows: In Section 2, we first classify the translation hyperovals in semifield planes into two types (a) and (b). Then we concentrate on hyperovals in Π(Kn), and we present a new family of transla- tion hyperovals of type (b) in it. Moreover, we give a complete list of translation hyperovals in Π(K5). In Section 3, instead of studying translation hyperovals in Kt Π( n), we equivalently look at a certain type of line hyperovals in its dual plane Ktd Ktd Π( n ). A new family of line hyperovals in Π( n ) and a complete list of them in Ktd Π( 5 ) are both presented. In Section 4, we also get a new family of translation Ktd Ktd hyperovals in Π( n ) and a complete list of them in Π( 5 ). Finally in Section 5, we turn to the hyperovals obtained by applying collineations on the translation hyperovals in semifield planes and the sizes of pairwise intersections of them. We prove that, for each hyperoval of type (a) in a semifield plane, there is a sym- metric design as well as two associated difference sets with the same parameters q2, q2/2+ q/2, q2/4+ q/2 , which are defined in two different abelian groups. 2. Hyperovals in Π(Kn) In [19, Corollary 2], it was proven that in every plane Π(P) coordinatised by a commutative (pre)semifield (P, +,⋆) of even order there is a translation oval, whose set of affine points is (1) (x, x ⋆ x): x F . { ∈ q} In the same article the authors refer to such an oval as the standard oval. Naturally, one can always complete the standard oval into a hyperoval s; for instance, by adding (0), ( ) as a carrier set. In [9], de Resmini, GhinelliO and Jungnickel used relative{ difference∞ } set to prove that in every commutative semifield plane, there is an oval, which is equivalent to the oval defined by (1). When P is a finite field, clearly s is the hyperoval comprising the points of a conic and its nucleus in PG(2, q), i.e.,O a regular hyperoval. As the presemifield Kn is commutative, if we replace ⋆ by its multiplication in (1), we get a hyperoval in Π(K ). ∗ Os n In this section, we proceed to show new hyperovals in Π(Kn). To consider the equivalence between different hyperovals, we need to know the automorphism group of the projective plane Π(Kn). As Hughes and Piper showed in [17, Lemma 8.4 and Theorem 8.6], the automorphism group of a semifield plane can be decomposed in the following way: S S Theorem 2.1. Let Π( ) be a semifield plane defined by a semifield . Let Γ(l∞, l∞) be its translation group and Γ(( ), l0) the group of shears (automorphisms fixing ∞ ( ) and l0).
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